Some new problems in additive combinatorics

Abstract

In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\dots ,a_n$ with adjacent sums $a_i+a_{i+1}$ (or differences $a_i-a_{i+1}$) pairwise distinct. For an odd prime power $q=2n+1>13$ with $q\ne 25$, we show that there is a circular permutation $(a_1,\dots ,a_n)$ of the elements of $S=\{a^2:a\in {\mathbb{F}}_q\setminus \{0\}\}$ such that $\{a_1+a_2,\dots ,a_{n-1}+a_n,a_n+a_1\}=S$, where ${\mathbb{F}}_q$ denotes the field of order $q$. For any finite subset $A$ of an additive torsion-free abelian group $G$ with $|A|=n>3$, we prove that there is a numbering $a_1,\dots ,a_n$ of the elements of $A$ such that $$a_1+2a_2,a_2+2a_3,\dots ,a_{n-1}+2a_n,a_n+2a_1$$ are pairwise distinct. We also pose 30 open conjectures for further research.

19 pages

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